REPORT RESUMES ED 016623 SE 003 950 INSTRUCTIONAL GUIDE FOR BASIC MATHEMATICS 1,GRADES 10 TO 12. BY- RICHMOND, RUTH KUSSMANN LOS ANGELES CITY SCHOOLS, CALIF. REPORT NUMBER X58 PUB DATE 66 EDRS PRICE MF -$0.25 HC-61.44 34P.
DESCRIPTORS- *CURRICULUM DEVELOPMENT,*MATHEMATICS, *SECONDARY SCHOOL MATHEMATICS, *TEACHING GUIDES,ARITHMETIC, COURSE CONTENT, GRADE 10, GRADE 11, GRADE 12,GEOMETRY, LOW ABILITY STUDENTS, SLOW LEARNERS, STUDENTCHARACTERISTICS, LOS ANGELES, CALIFORNIA,
THIS INSTRUCTIONAL GUIDE FORMATHEMATICS 1 OUTLINES CONTENT AND PROVIDES TEACHINGSUGGESTIONS FOR A FOUNDATION COURSE FOR THE SLOW LEARNER IN THESENIOR HIGH SCHOOL. CONSIDERATION HAS BEEN GIVEN IN THEPREPARATION OF THIS DOCUMENT TO THE STUDENT'S INTERESTLEVELS AND HIS ABILITY TO LEARN. THE GUIDE'S PURPOSE IS TOENABLE THE STUDENTS TO UNDERSTAND AND APPLY THE FUNDAMENTALMATHEMATICAL ALGORITHMS AND TO ACHIEVE SUCCESS ANDENJOYMENT IN WORKING WITH MATHEMATICS. THE CONTENT OF EACH UNITINCLUDES (1) DEVELOPMENT OF THE UNIT, (2)SUGGESTED TEACHING PROCEDURES, AND (3) STUDENT EVALUATION.THE MAJOR PORTION OF THE MATERIAL IS DEVOTED TO THE FUNDAMENTALOPERATIONS WITH WHOLE NUMBERS. IDENTIFYING AND CLASSIFYING ELEMENTARYGEOMETRIC FIGURES ARE ALSO INCLUDED. (RP) WELFARE U.S. DEPARTMENT OFHEALTH, EDUCATION & OFFICE OF EDUCATION
RECEIVED FROM THE THIS DOCUMENT HAS BEENREPRODUCED EXACTLY AS POINTS OF VIEW OROPINIONS PERSON OR ORGANIZATIONORIGINATING IT. OF EDUCATION STATED DO NOT NECESSARILYREPRESENT OFFICIAL OFFICE
POSITION OR POLICY.
INSTRUCTIONAL GUIDE
BASICMATHEMATICS I GRADES 10 to 12
LOS ANGELES CITY SCHOOLS Division of Instructional Services Curriculum Branch Publication No. X-58 1966 FOREWORD
This Instructional Guide for Basic Mathematics 1 outlines content and provides teaching suggestions for a foundation course for the slow learner in the senior high school. Consideration has been given to the student's interest levels and his ability to learn.The guide's purpose is to enable the students to understand and apply the funda- mental mathematical algorithms and to achieve success and enjoyment in working with mathematics. It is anticipated that the teacher, too, will find a challenge in experimenting with a new development of basic topics and that he will experience the reward of observing real progress (in the part of class members.
It is suggested that the teacher remind himself of the importance of treating each student as a worthy individual, of providing for frequent changes of pace, and of being generous with praise for accomplishment and for effort. The slower learner needs work which is within his comprehension, yet which encourages exploration and discovery. Be needs assignments which are long enough to provide real learning experiences, but brief enough to encourage their suc- cessful completion..
Teachers who approach the teaching of Basic Mathemardcs 1 with enthusiasm and interest will find the experience to be rewarding. The challenge of the general mathematics class calls forth real teaching skills and a thorough understanding of the subject. Most of all, the Basic Mathematics class stimulates the teacher's best efforts in reducing to the simplest of terms the basic ideas and the more sophisticated concepts which are developed. 1
ACKNOWLEDGMENTS
Appreciation is expressed to the following committee members who assisted in developing the outline, contributed teaching suggestions, and reviewed the manuscript for this publication:
Roger Kent Anderson San Fernando Senior High School Marjorie Tanton McKesson Cleveland High School Max R. Wainwright, Jr. Manual Arts High School
Appreciation also is conveyed to KEITH F. SMITH, Supervisor, Division of Secondary Education, for his contributions to the work of the committee through participation in discussions and review of the materials which were prepared.
To RICHARD DENHOLM, Coordinator of Mathematics and Science, Orange County Schools, special acknowledgment is due. Dr. Denhoim, who served as a special NDEA consultant for the project, brought to the task a wealth of knowledge, experience, and writing ability.His good advice, numerous specific suggestions, and critical reading of the manuscript were of inestimable assistance.
To RUTH KUSSMANN RICHMOND, who served as consultant for the project, the Curriculum Branch is especially indebted for developing, organizing, and writing the guide.Mrs. Richmond's breadth of teaching experience in secondary schools and in the junior college and her thorough knowledge of the problems of the slow learner provided an invaluable background for the assignment.
MARIAN C. HERRICK Supervisor, Secondary Mathematics Curriculum Branch
MARY LOUISE JONES Secondary Curriculum Coordinator
AVERIiPL M. CHAPMAN APPROVED: Administrator of Curriculum
ROBERT E. KELLY Associate Superintendent Division of Secondary Education
EVERETT CHAFFEE Associate Superintendent Division of Instructional Services
-iv- TABLE OF CONTENTS
LSO
FOREWORD. 41 ap iii
ACKNOWLEDGMENTS iv
OBJECTIVES OF BASIC MATHEMATICS 1 vi
UNIT I. MATHEMATICS IN OUR MODERN WORLD 1 - Using mathematics in daily living - Recognizing mathematics in the student's world
UNIT II. THE SET OF WHOLE NUMBERS:ORDER AND RELATIONSHIPS 5 - Recognizing, relating, and using whole numbers - Measuring lengths
UNIT III. ADDITION:OPERATION WITH WHOLE NUMBERS 9 - Understanding addition of whole numbers - Developing order concepts - Measurement
UNIT IV. SUBTRACTION:OPERATION WITH WHOLE NUMBERS 13 - Understanding subtraction as the opposite of addition - Using subtraction to solve practical problems
UNIT V.NUMBER CONCEPTS 17 - Developing concepts of numbers greater than 10 - Using expanded notation
UNITVI.MULTIPLICATION: OPERATION WITH WHOLE NUMBERS 23 - Understanding multiplication as repeated addition - Developing multiplication tables - Studying areas
UNIT VII. DIVISION: OPERATION WITH WHOLE NUMBERS 29 - Understanding division as repeated subtraction - Using division to solve practical problems OBJECTIVES OF BASIC MATHEMATICS 1
The course in Basic Mathematics 1 is planned to provide instruction which is suitable for the slow learner in senior high school whose ability in general mathematics is considerabley below expectancy for his grade.
The course is intended to enable the student to achieve:
Vocational competence, as he develops the ability to
- -Followdirections accurately. - -Add, subtract, multiply, and divide whole numbers. --Perform the basic operations of arithmetic with accuracy.
- -Judgethe reasonableness of his solutions to problems. - -Copy corectly numerals, symbols, and simplegeometric figures.
Consumer effectiveness, as he learns to Purchase merchandise and other items wisely. - -Plan and manage a budget and handle financialtransactions with a bank intelligently.
- -Estimatequantities, distances, areas, and weights with reasonable accuracy. - -Read advertisements with understanding todetermine the amounts actually required for the purchase of the adver- tised items.
Responsible community: membership, as he grows in ability to --Understand the importance of integrity and develop a personal commitment to high standards of honesty. BASIC MATHEMATICS 1 UNIT I. MATHEMATICS IN OUR MODERN WORLD 7 Teaching Days
110..111111111.11=1=MIMEN
A. Develoment of the Unit
1. Develop an awareness of the importance of mathematics inour daily lives. a. Discuss the mathematics which the students need touse as workers and citizens. b. Explore the many ways in which mathematics is basic to human activities. c. Point out the contribution which mathematics has made in the space program.
2. Discuss the students need to understand mathematics in order to takea significant place in society. a. Point out how mathematics is related to Money Purchasing and paying bills Housekeeping Owning a car
3. Use items from newspapers to illustrate the widespreaduse of mathematics. a. Point out the necessity of reading a newspaper discerningly. b. Help students to discover articles and ads which involve mathematics and mathematical symbolisms.
4. Introduce the subject of taxes. a. Discuss the reasons why taxes are necessary. b. Help students to learn the methods for computing and collecting taxes. c. Discuss types of taxes collected. Local State Federal
5. Emphasize the importance of learning about measurement. a. Time b. Temperature c. Age d. Lengths and distances e. Area lmILALIlmaggitisms IN OUR MODERN WORLD
B. Svgested Procedures
The teacher may find it usefulto: students to list ways in which they usemathematics every day.
--Discuss ways in which everyoneis called upon to use mathematics.
--Develop recognition of the needfor mathematics by studying some of its history.
--Introduce the topic of spaceexplorntion, and point out the important role of mathematics.
-.Plan with class an imaginary tripto the store to purchase groceries.
--Initiate a discussion about owning a car(purchase price, mileage, gas consumption,insurance, payments, andresponsibilities).
--Bring newspapers to school anddistribute copies to class members.
--Ask students to cut out articlesr4nd ads in which mathematical concepts are used. Discuss their content.
--Point out the need for reading newspapersintelligently.
--Introduce a discussion about theneed for taxes.
a. Discuss the establishment of a smallwestern town.
b. Show the need for roads, schools, afire department.
c. Show the need for a sheriff, and ajail. d. Discuss possible ways of collectingtaxes.
- -Askstudents to compute their ages in years,months, and days.
--Discuss the day's weather report,and review the use of the thermometer.
-Z- BASIC MATHEMATICS
B. Suggested Procedures (contd)
--Conduct oral drill in telling time.Ask class members toexpress time in differentways, such as 9:10 half after 6 half past 5 3:45 quarter to 8 quarter past 1
--Discuss the purposes anduses of measurement.
C. Evaluation
It is expected that the students willbe able to demonstrate that they have learned:
1. The need for studying. mathematics.
2. To read a newspaper discerningly.
3. To interpret more successfully the worldaround them and recognize its dependence upon mathematics.
4. To recognize the need for taxes.
5. To appreciate the need for measurement. BASIC MATHEMATICS 1 UNIT II. THE SET OF WHOLE NUMBERS:ORDER AND RELATIONSHIPS 14 Teaching Days
A. Development of the Unit
1. Introduce the set of whole numbers. a. Develop the concept that a whole numbercan be expressed in many ways.
b. Show that most fractions and decimalsare not whole numbers. c. Describe the set of whole numbers as a set which includes 0 and the counting numbers. d. Lead students to discover that in the set of whole numbers, there is no largest whole number.
2. Develop the number line as a device to represent number ideas. a. Introduce the ',et of whole numbers on the number line. b. Show how to construct a number line.
3. Interpret number as a property of setsor collections. a. Relate number property of sets to points on the number line. b. Describe equivalent sets as sents which contain the same number of members and which have an identical number property.
4. Present number patterns on the number line. a. Develop a variety of number sequences and sets. b. Develop the ability to complete open numbersequences.
S. Develop the order property of the set of whole numbers. a. Relate the order of whole numbers to the number line.
b. Develop an understanding of related terms, suchas: "is equal to" "is greater than" "is less than" UNIT II THE SET OF WHOLENUMBERS:ORDER AND RELATIONSHIPS
A. Development of the Unit (contd)
5. c. Develop an understandingof symbolism.
mis means"equal to" > means "is greaterthan" < means "is less thew"
6. Introduce the use of wholenumbers in problem solving. a. Relate an understanding ofthe order of whole numbers to problem situations.
b. Develop problem solvingskills through theuse of problems written by students.
7. Introduce basic geometricconcepts. a. Develop the concept ofa point. b. Develop the concept ofa line as a set of points.
c. Present the basic geometricshapes: square, rectangle, triangle, circle, parallelogram,polygon. d. Introduce the concept ofan angle.
8. Present the fundamentalsof measurement. a. Develop the concept of linearmeasurement. b. Relate the concepts of "isgreater than" and "is less than" to measurement.
B. Suggested Procedures
The teacher may find it usefulto:
--Provide students with activitieswhich will illustrate thatthere are many ways of expressing each whole number
3 1 + 1 + 1 III 5 - 2 2 + 1 etc.
--Ask students to write theset of whole numbers beginningwith 0. Explain what the symbol ". . ." means after the first fewnumerals. For example: { 0, 1, 2, .1'
--Explain that there isno largest counting number. Demonstrate by taking a very large number andadding 1 to it to producean even larger number. 5,000,000 + 1 = 5,000,001.
--Show that most fractions anddecimal fractionsare not whole nuMbers. 1 .5 2."2# 1. BASIC MATHEMATICS 1
B. Aggested Procedures (contd)
--Illustrate concepts ofa number line.
Show students that to makea number line, a unit of length is first established.
After the unit of length isestablished, point out that theposition of each number is determined insuch a way thatany two adjacent points representing whole numbersare the same distance apart. Explain that a number linehas no end points. The arrows at the ends of the number line imply thisprinciple.
--Instruct students to relate the numberproperty of a collection of objects in a set to the numeralson the number line.
--Help students to generalize that (X X X) and19?)have the same number property as wellas the same number of elements andare therefore called equivalent sets while (X XX X) and (5° are not equivalent sets.
Show by simple illustrationsthat number lines may havemany different patterns.
a 0 2 4 6 8 10
o 1 3 ;3
--Introduce a variety of number patternsfor students to explore.
Ask students to completemany partial number scquences.
{2,4, ,16, ,128.}
--Use the number line to show thateach point represents a number less than any point to its right andgreater than any point to its left.
--Conduct drill in theuse of "is equal to," "is greater than," and "is less than."
5 is less than 6 i, 3 34.1 is equal to 2+2 4 0 6 is greater than 5 0 4 UNIT THE SET OF WHOLE NUMBERS: ORDER AND RELATIONSHIPS
B. Suggested Procedures (contd)
--Introduce symbols > "is greater than" and< 'is less than." Help students to generalize that the arrowalways points to the smaller number. 3>2 2 <3
--Provide drill in the use of these symbols. 3 e 4 5
4 --Assign simple word problems that willillustrate each of the concepts developed in this unit. --Present basic concepts of geometry stressingthe concepts developed in this unit.
- -Startwith a point, and then show that aline is a set of points. Introduce the basic shapes which arepresent in the physical world. Ask students to answer simple questionsabout the number of sides and angles in the various shapes.Develop the idea of "greater than" or "less than" as relatedto sides and angles. --Introduce students to measurement through the useof a straight edge. (It is recommended that the straight edge bemarked only with inch, one-half inch, and one-fourth inchmarkings.) --Present many examples to illustrate howmeasurement is performed. Measure sides of triangles, quadrilaterals,rectangles, squares, parallelograms, and polygons.Show students how the concept of "is greater than" or "is less than" isapplied to measurement.
C. Evaluation
It is expected that the student willbe able to demonstrate that he has learned to:
1. Distinguish between whole numbers,fractions, and decimal fractions.
2. Make a number line, identify patterns on anumber line, and indicate what numbers are necessary to complete a sequenceof numbers correctly.
3. Use the concept of a number line todetermine which numbers are "greater than" or"less than" other numbers.
4. Solve simple word problems relating to theconcept of whole numbers and the order of whole numbers.
5. Use effectively the ideas of point, line,and common geometric shapes.
6. Measure lines and apply the concepts of"greater than"or "less than" to the measured lines. BASIC MATHEMATICS 1 UNIT III. ADDITION: OPERATION WITH WHOLE NUMBERS 12 Teaching Days
A. Development of the Unit
1. Introduce addition as the joining of sets of objects that have no members in common. a. Develop the commutative property for joining sets.
b. Develop the associative property for joining sets.
2. Introduce addition, using the number line.
a. Develop an understanding of whole numbers as represented by unit distances from zero on the number line. b. Develop addition as the combining of number line distances representing whole numbers.
3. Explore the commutative property for addition of whole numbers.
a. Develop the concept that the order of addends does not affect the sum. b. Emphasize the commutative property in both horizontal and vertical addition.
4. Introduce the concept of zero as applied to addition.
a. Develop an understanding of zero as the identity element for addition.
b. Emphasize the use of the identity element in addition problems.
5. Explore the associative property for addition of whole numbers.
a. Develop the concept that, when three or more numbers are to be added, the sum is independent of the grouping. b. Emphasize the associative property by both horizontal and vertical addition.
6. Introduce practical problems which are solved by the use of addition.
7. Help students to increase their understanding of basic geometric concepts.
a. Develop the concept of line segment. b. Develop the concept of perimeter.
8. Help students to understand the fundamentals of measurement.
a. Develop the addition of measures of lines.
b. Apply the concepts of addition of line segments to perimeters.
c. Develop the concept of area. UNIT III.ADDITION:OPERATION WITH WHOLE NUMBERS
B. Suggested Procedures (contd)
The teacher may find it useful to: --Provide student activities to helpstudents understand the operation of addition through the joiningof two sets.
B
-I- 2 5
--Develop the understanding of unionof disjoint sets to includethree or more sets. --Instruct students to representwhole numbers on the number line as distances from zero.
2
I I 4
3
0 1 5
--Ask students to demonstrateaddition facts on the number line.
3 2 3+2=5
0
--Provide students with a series ofrelated problems leading to the discovery of the commutative property ofaddition.
2+3= 5-1-1 = 3+2= 1+5 =
2 3 5 1 +3 +2 +1 +5 BASIC MATHEMATICS 1
B. Suggested Procedures (contd) - -Ask students to solve related problems on the number line.
2 3 2 +3 = 5
I I I I I I
1 1 2 3 1 8 9 10
3 2 3 + 2 = 5
--Supply problems in which zero is introducedas an addend.
5+0= 8+0=0 0 +5= 0+8 =
5 0 8 0 +0 +5 +0 +8
- -Present problems that will help students to discover the associative property of addition.
(2 + 4+4 = 1111 2 2
2 +(3 + = [÷3 4 4c+ 43 El El
- -Develop the associative property in the use of the number line. ( 2 + 3 ) + 4 (2+ 3)+4
0 10 11
2 +4) 2 +0 + /0=
--Assign a series of simple word problems which require studentsto employ concepts learned in this unit. UNITIII. ADDITION:OPERATION WITH WHOLE NUMBERS
B. Suggested Procedures (contd) --Encourage students to write and solve their own problems using concepts developed in this unit.
--Instruct students to use a straight edge to draw line segments.
--Ask students to distinguish line segments from lines.
- -Helpstudents to discover that the perimeter of any plane figure is the distance around it. 4
- -Helpstudents to understand the addition of denominate numbers and the measures of line segments that make up perimeters or plane shapes.
- -Introducestudents to the concept of area by counting unit squares,
C. Evaluation
It is expected that the students will be able to demonstrate that they have learned to
1. Recognize the number property from the number of objects in a set and add the elements of two or more sets.
2. Demonstrate the commutative property and the associative property for addition on the number line.
3. Understand that zero is the identity element for addition.
4. Solve simple word problems, using addition of whole numbers in situations.
5. Write and solve simple word problems.
6. Add the measures of line segments and compute the measure of the perimeters of some common figures.
7. Apply concepts of area and its measure to the solution of problems.
-12- BASIC MATHEMATICS 1 UNIT IV. SUBTRACTION:OPERATION WITH WHOLE NUMBERS 10 Teaching Days
A. Development of the Unit
1. Introduce subtraction as the opposite of addition. a. Develop subtraction as a separation of sets.
b. Show the relationship of addition and subtractionfacts on the number line.
2. Present subtraction problems in both horizontal andvertical form. a. Relate pairs of equations. b. Relate pairs of problems in vertical form. c. Interpret subtraction as discovering the missing addend.
3. Introduce the concept of zero in subtraction.
a. Develop the idea that subtracting 0 from any number results in the same number. b. Develop the idea that subtractingany number from itself produces zero.
4. Introduce the vocabulary used in performing subtraction. a. Present definitions of subtrahend, minuend, difference, and missing addend. b. Illustrate subtrahend, minuend, difference, and missing addend.
5. Introduce problems in which the subtractionprocess is utilized.
6. Extend understandings of basic concepts of geometry. a. Review basic concepts of geometry. b. Present examples of a plane, acurve, a closed curve, and interior and exterior regions.
7. Help students to understand measurement. a. Redevelop the idea of area by counting and subtracting. b. Provide practice in subtraction of themeasures of line segments. UNIT IV. SUBTRACTION:OPERATICS WITH WHOLENUMBERS B. Suggested Procedures
The teacher may find ituseful to:
-.Provide activities toshow students therelationship of set separation to subtraction.
X X X XXX X X X X X
--Ask students to make up their own equations from setillus- trations to show thisrelationship.
6+2= 8-2= 8- 6=
--Instruct students todemonstrate subtractionfacts on the number line.
4 5 4+5=9
0 1 2 3 5 6 7 8 9-5=4 5
--Ask students to showthe relationship betweenaddition and subtraction by writing addition sentences fromsubtraction equations.
5- 3 = 3+ =5 BASIC MATHEMATICS 1
B. Suggested Procedures (contd)
--Provide subtractionexercises to be solved inboth the vertical and horizontal forms.
7 1 5 + 2 = 7 MM. 7 -0= 2 7 2 5 1- = 5
--Introduce zeroas the subtrahend in subtractionproblems. 5 5 5 + 0 = 5 +0 -0 5 0 = 5 5 5
Develop the idea thatn 0= n and that n- n = 0.
--Develop the meaningsof terms suchas subtrahend, minuend, difference, and addend. --Assign students a series of word problems,using concepts learned in this unitas well as concepts learnedin previous units. - -Encourage students to makeup problems of thairown involving subtraction. --Review and redevelop geometric ideas previouslypresented. - -Develop the concept of a plane by usingas examples a desk top, floor, ceiling, and otherflat surfaces. - -Help students to interpret a curve as a set of points whichcan be represented bya pencil drawing made withoutlifting the pencil from thepage.
- -Ask them to mark with an X thecurves in the drawings below. CAD
- -Point out that a closed curve isa figure formed when aperson places a pencil point on a paper and moves it around andends at the point where hebegan. UNIT IV. SUBTRACTION:OPERATION WITH WHOLE NUMBERS
B.Suggested Procedures (contd)
--Conduct drill exercises to emphasize the meaning of interior and exterior regions of closed curves.
--Provide activities that will help class members to developthe idea of interior and exterior as applied to regions in geometric figures.
--Assist students in discovering the application of subtraction in the measurement of distances.
C. Evaluation
It is expected that the student will be able to demonstrate that he has learned to
1. Relate addition facts to subtraction facts anduse them in solving practical problems.
2. Show subtraction facts on the number line.
3. Complete open number sentences, using addition and subtraction facts.
4. Understand the use of zero as an identity element for subtraction.
5. Apply such terms as subtrahend, minuend, difference, and addend correctly.
6. Interpret applications of plane figures in the real world.
7. Recognize curves and closed curves.
1.0,b0a. BASIC MATHEMATICS 1 UNIT V. NUMBER CONCEPTS 14 Teaching Days
A.Development of the Unit
1. Introduce number-numeral relationships.
a. Show that there are many ways of writinga number. b. Develop an understanding that numeralsare man-made symbols to represent number ideas.
2. Introduce expanded notation. a. Develop the concept of numerals havingmore than one digit. b. Relate expanded notation to place value. c. Develop the idea of regrouping.
3. Relate ideas of addition and subtraction to expanded notation. a. Show alltions of whole numbers with and without regrouping. b. Show subtractions of whole numbers with and without regrouping.
4. Introduce problems for students to solve in which expanded notation with and without regrouping is used.
5. Extend the understanding ur geometric concepts. a. Explore the different types of quadrilaterals and triangles. b. Develop working definitions of quadrilateral and triangle. c. Develop understandings of properties of angles.
6. Continue the study of measurement. a. Introduce the use of the protractor for angle measurement. b. Use angle measurement to classify angles Straight angle Right angle Acute angle Obtuse angle UNIT a9 NUMBER CONCEPTS
B. suggested icedures
The teacher may find ithelpful to:
- -Assign activities in which students willdiscover differentways of naming numbers.
1
1 0 6 + 1 8 1 SEVEN 1 1 + 5 9 2 V i i
--Review cldbinations ofnumbers that add to 10.
= 10
--Develop new symboli torepresent numbers from 0 to10.
0 1 2 3 4 5 6 7 8 9 10
- -Provide drill exercises in whichstudents will use thesenew symbols.
ED+
- -Extend the idea of naming numbers to include numbers largerthan 10. 9 + 6 as 15 8 + 7 = 15 10 +5 =15
--Rename numbers to emphasizeplace value.
11= 10 + 7 135 = 100 + 30 + 5 26 = + 6 26 = 20 + 6 456 = + 50 + 26 = + TENS+., ONES 32 = + 2 .21 300 + 40 + 5 2 TENS + ONES
- -Introduce regrouping of numbers. 26 = 20 + 6 26 =(10 + 10)+ 6 26 = 10 + (10 + 6) 26 In10 + 16 BASIC MATHEMATICS 1
B. S2211Droseuestecdures (contd)
--Assign practice in regrouping numbers.
324=300+ 20+ 4= + +4 324=300+ (10+ 10)+ 4 = + _+ 10)+ 4 324=300+ 10+ (10+ 4) =__,__+ + (10+ 4) 324=300+ 10+ 14= + 10+
Develop an understanding of addition,using the conventional vertical form and using expanded notation.
23 20 + 3 + 42 40 +2 65 60 + 2 = 65
--Extend addition to include regrouping.
28 20 + 8 +45 40 + 5 73 60 + 13 = 70 + 3 = 73
--Develop an understanding of subtraction,using the conventional vertical form and using expanded notation. 35 30+5 _23 -(20 + 3) 12 10 + 2 = 12
--Extend subtraction to include regrouping. 23 20+ 3 10 + 13 - 18 -(10 +8) =-(10 + 8) 5 5
--Provide students with simple word problemsusing expanded notation and regrouping. UNIT V. NUMBER CONCEPTS
B. Suggested Procedures (contd)
- -Provide activities requiring students to identify quadrilaterals. - -Instruct class members to mark with an X the figures that are quadrilaterals: vs=1 arIC)0 - -Discuss quadrilaterals to develop an understanding of their general properties.
Ask the class, "What do you notice about each?"
--Develop an understanding of an angle as the union of two rays.
--Develop an understanding of interior and exterior regions of angles.
EXTERIOR
- -Relate the concepts of "greater than" and "less than" to angles. - -Discuss the protractor and its composition, and use. - -Provide activities requiring students to use the protractor to measure angles. - -Help students to develop skills in the classification of angles by size. BASIC MATHEMATICS 1
C. Evaluation
It is expected that the student will be able to demonstrate that he has learned to:
1. Express numbers in many different ways.
2. Express number ideas in the conventional form and in expanded notation.
3. Recognize the positional values of digits in numerals.
4. Apply the concept of expanded notation to solve addition and subtraction problems.
5. Solve problems involving addition and subtraction with and without regrouping.
6. Differentiate between quadrilaterals and other polygons.
Differentiate between triangles and other polygons.
8. Classify quadrilaterals, triangles, and angles.
9. Use the protractor to measure angles.
-21- BASIC MATHEMATICS 1 UNIT VI.MULTIPLICATION:OPERATION WITH WHOLE NUMBERS 16 Teaching Days
A. Development of the Unit
1. Introduce multiplication by using arrays of collections of objects.
a. Develop the commutative property by using arrays.
b. Introduce multiplication by using arrays having equal numbers of members.
2. Introduce multiplication by the use of the number line.
a. Develop an understanding of whole numbers as distances on a number line.
b. Develop multiplication as repeated marking off of a unit distance along a number line.
3. Introduce the commutative property of multiplication.
a. Develop the concept that the order of factors does not affect the product.
b. Demonstrate the commutative property of multiplication on a number line.
4. Introduce the associative property of multiplication.
5. Introduce the concepts of zero and one as applied to multiplication.
a. Develop the concept of one as the identity element for multiplication. b. Develop the concept that the product of zero and any number is zero.
6. Apply concepts of expanded notation and place value in multiplication.
7. Introduce problems for students in which the use of multiplication is involved.
8. Develop further understandings of geometric concepts. Ask students to use concepts of properties of angles to construct Straight angles Acute angles Right angles Obtuse angles
9. Extend and redevelop the concepts of measurement.
a. Develop the concept of area.
b. Develop the concept that the sum of the angles of a quadrilateral is 360 degrees.
c. Explore the concept that the sum of the angles of a triangle is equal to 180 degrees.
d. Assist students in developing an intuitive understanding of areas of squares and rectangles.
-23- UNIT VI.MULTIPLICATION: OPERATION WITH WHOLE NUMBERS
B. Suggested Procedures
The teacher may find it useful to: --Use rectangular arrays to interpret multiplication.
4{XXXXXX 3 XXXX XXX 'CX XX XX XXXX XXX XX xx XX 4 3 4 + 4 + 4 - 12 OR 4=12 4x3=12 3)(4=12
2 2 2
I I I ir 1 I I I 6 ; 0 0 1 2 3 4 5 1 2 3 4 8 2+2+2= 6 3 + 3 = 6 3 X 2 = 6 2 X 3 = 6
--Develop the commutative property by showing pairsof related equations
1x3= 3 2x4=8 4X1 =4 18= 6 x 3 3 x 1= 3 4x2 =8 1x4=4 18=3* 6 6=2x 3 12=3)(4 3x0=0 18= 9%2 6= 3x2 12=4x3 0x3=0 18= 2x9 BASIC MATHEMATICS
B. Suggested Procedures (contd)
--Develop the associative property by showing pairs of related equations on the number line.
(1x2)x3= 1X2 1x2 1x2 1X (2x 3)'
(1X 3) X 0 = 3
(1x 2)X 3= 1 X (3 X 0).=
--Introduce the multiplication table (matrix) through 9 x 9)
X 0 1 2 3 4 5 6 7 8 9
1
1
2 .1'4 v 3 121 4
5 , 6
7
8
9 --Ask class what pairs of factors give the product in the following
X 21 FACTOR x FACTOR= 21
3 XI:=21 FACTOR x = PRODUCT E:IX 7 = 21 x FACTOR = PRODUCT
--Introduce multiplication where zero is a factor 5 X 0 =
5X 0 UNIT VI.MULTIPLICATION: OPERATION WITH WHOLE NUMBERS
B. Suggested Procedures (contd)
--Combine the commutative and associative propertiesof multiplication. --Provide practice writing exercises in numbersin expanded notation. 115 = 100 +10 + 5
- -Ask students to build multiplication tables (matrices) for multiples of ten.
X 1 10 100 1000
1 1 10 100 1000 20 30
2 2 20 200 2000 1 1 20 30 40
3 3 30 300 3000 2 20 40 60 80 4 4 40 400 4000 3 30 60 90 120
--Help students to relate multiplication factsto expanded notation.
23)c 2 = 20+ 3 23 x2 x2 40 + 6 =46 46
132 x 3 =100 + 30 + 2 132 x3 x3 300 + 90 + 6 = 396 396 - -Introduce problems for students in which multiplication must be used. - -Explain methods of constructing straight angles, right angles, acute angles, and obtuse angles.
- -Ask students to construct quadrilaterals. Instruct them to measure angles and to find the sums.
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80e BASIC MATHEMATICS 1
B. Suggested Procedures (contd)
--Ask students to construct triangles, to measure their angles, and to find the sum of the angles.
--Redevelop the idea of area and correlate the counting of squares to the concept of length and width as it relates to finding areas of rectangles and squares,
C. Evaluation
It is expected that students will be able to demonstrate that they have learned to
1. Use effectively the idea of multiplication as repeated addition.
2. Interpret multiplication by means of arrays and a number line model.
3. Make use of the commutative and associative properties of multiplication.
4. Complete multiplication matrices (tables) for whole numbers through 9 and for decades. (As 20, 30, 40.)
5. Recognize that one and zero have special properties related to multiplication.
6. Solve multiplication problems written in expanded notation and in the conventionrl vertical form.
7. Solve simple word problems using multiplication.
8. Construct special angles.
9. Measure angles of quadrilaterals and triangles and observe a geometric fact about the sum of the angles of each.
10. Use the formulas for the area of squares and rectangles. BASIC MATHEMATICS 1 UNIT VII. DIVISION:OPERATION WITH WHOLE NUMBERS 15 Teaching Days
A. Development of the Unit
1. Introduce division as the opposite of multiplication.
a. Present division as set partitioning. b. Present related division and multiplication facts together.
2. Present division as repeated subtraction.
a. Develop division on the number line. b. Show that division can be accomplished by repeated subtraction.
3. Develop division as location of the missing factor.
4. Develop the properties of the operation of division. a. Develop the concept that division of any number by one produces that same number.
b. Develop the concept that dividing zero by any number produces zero.
c. Develop the concept that division by zero is undefined.
d. Develop the concept that division is not commutative or associative.
5. Introduce vocabulary used in division.
a. Present definitions of dividend, divisor, and quotient.
b. Identify dividend, divisor, and quotient in various types of problems.
6. Introduce geometric concepts that include properties of the circle.
a. Help students to interpret the relationship between the diameter and the radius of a circle.
b. Develop the concepts of circumference and arc.
c. Introduce chord and tangent.
d. Discuss the concept of the center of a circle.
e. Develop the idea that a circle is a closed curve dividing a plane with three sets of points.
7. Continue the study of measurement.
a. Show the area of a circle by counting unit squares. b. Assist students in applying the concepts of addition, subtraction, multiplication, and division to denominate numbers.
-29- UNIT VII. DIVISION:OPERATION WITH WHOLENUMBERS
B. Suggested Procedures
The teacher may find ituseful to: --Interpret division as the opposite of multiplicationby using sets of physical objects andarrays. X X X X X 2X32=6 X X 4X2 =8 2 4 X X X 64.2=3 X X 8+2=4 3 X X --Use pairs of related equations to emphasizethat division is the opposite of multiplication. 2x =8 8 +2= --Use the concept that Factor x Factor= Productto develop concept thatProduct- Factor = Factor.
3X 2= 6 Factor x Factor= Product 6 +2 =3 6 Product 4- Factor= Factor
--Develop activitiesto demonstrate that a. Division of any number by one producesthat same number. 3:-1 =3 5 + 1 = 5 3+0=3 11+1= 5 b4 Division of zero by any number produceszero. 0 + 5 = 0 0+ 6= 0 c. Division of any numberby zero is undefined.
d. Division is neithercommutative nor associative. True or False 4+ 2= 2+4 6+3= 3+6 8+4+2=8 + 4+2
--Introduce activities tobuild understandingof terns.
a. Divisor b. Dividend c. Quotient BASIC MATHEMATICS
B. Suggested Procedures (contd)
- -Askstudents to construct a circle and to draw several diameters.
- -Help classmembers to develop intuitive definitions of diameter and radius. Establish the relationship between radius and diameter. --Instruct students to use string and tape measures to estimate circumference.
- -Assiststudents in developing an intuitive understanding of the relationship between diameter and circumference.
- -Point out tostudents the names for other parts of a circle. --Review units of linear measure and area measure. --Use the counting of unit squares in rectangles and squares to assist students in discovering formulas for area. - -Instruct classmembers to use squared paper to construct a circle and estimate the area. alley may use inches or any other unit of measure.)
- -Provide practiceexercises in subtracting, multiplying, and dividing denominate numbers.
3 ft. 2 in. 35° 25' o + 2 ft. ,5 _in. 10 20' 5 ft. 7 in. 47° 45' UNIT VII. DIVISION:OPERATION WITH WHOLE NUMBERS BASIC MATHEMATICS 1 C. Evaluation
It is expected that students will be ableto demonstrate thatthey have learned to:
1. Relate division andmultiplication.
2. Demonstrate that divisionof any number byone produces that same number.
3. Demonstrate that dividingzero by any number produces zero, but that division byzero is undefined.
4. Use the idea that Factorx Factor = Product and relate this to the fact thatthe Product + Factor= Factor.
5. Recognize that divisionis not commutativenor associative.
6. Know the vocabulary associated with theprocess of division.
7. Know the vocabulary relatedto the use ofa circle.
8. Develop intuitivelyformulas for computingthe areas of squares and rectangles.
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